Right fibrations over $N(\Delta)$ as complete Segal spaces

Question

Is there a Bousfield localization of the model category for right fibrations over $N(\Delta)$ that is Quillen equivalent to the model category for complete Segal spaces?

Answer 1

Yes.

One can perform the left Bousfield localization with respect to the maps [m] ⊔_[0] [n] → [m+n] and the map E → [0].

Here [m] denotes the representable fibration corresponding to the m-simplex and ⊔ denotes the homotopy pushout of fibrations.

In the latter map the source object E encodes the free groupoid on one arrow; it can be defined as the nerve of this groupoid, interpreted as a simplicial object in (discrete) simplicial sets.

Locality with respect to the first class of maps ensure the Segal condition, i.e., X_{m+n} → X_m ×_{X_0} X_n is an equivalence. Locality with respect to the second map ensures the completeness condition, i.e., X_0 → X_inv is an equivalence.